Issue 49

O. Naimark, Frattura ed Integrità Strutturale, 49 (2019) 272-281; DOI: 10.3221/IGF-ESIS.49.27 276 where p  are the kinetic coefficients. Kinetic equations (7) and the equation for the total deformation ˆ C p     ( ˆ C is the component of the elastic compliance tensor) represent the constitutive equations of materials with defects. Material responses include the generation of characteristic collective modes – the autosolitary waves in the range of * c      and the “blow-up” dissipative structure in the range 1 c     . The generation of these collective modes under the loading provides the defect induced mechanisms of structural (ductile) relaxation in the range * c      and specific mechanisms of damage localization on the set of spatial scales with the blow-up defect growth kinetics. The “blow-up” damage localization kinetics follows to the self-similar solution [21]     p g t f   , H x L   , ( ) (1 ) m c g t G t     , (8) and can be considered as the precursor of crack nucleation. The parameters in (8) are: c  is the so-called "peak time" ( p  at c t   for the self-similar profile   f  of defects localized on the scale H L ), 0, 0 G m   are the parameters of non-linearity, which characterize the free energy release rate for c    . The blow-up self-similar solution (8) describes damage kinetics for , c c t p p    (Fig.2) on the set of spatial scales , 1, 2,... H c L kL k K   , where c L and H L corresponds to the so-called “simple” and “complex” blow-up dissipative structures. The scales c L represent natural measure of the quantization length in the process zone providing the variety of the crack paths in the presence of two singularities: intermediate asymptotic solution for stress distribution at the crack tip area and the blow-up damage localization kinetics in the process zone. 0 a p p   c     c  c c p 1 p 2  1 F p p p p    c 1 c 2 1 a b Figure 2 : a. Nonlinear responses of material on defect density p in different ranges of structural-scaling parameter  ; b. Free energy metastability for 1 c     [19]. D UALITY OF SINGULARITIES . THE TCD LENGTHS he existence of two singularities related to the stress field at the crack tip (4) and blow-up kinetics of damage localization (8) represents the physical basis for the interpretation of phenomenological assumption of the Theory of Critical Distances. The free energy metastability of solid with defects and corresponding free energy release explain the conception of the Finite Fracture Mechanics in the presence of the finite amplitude energy barrier. The out-of- equilibrium system “solid with defects” realizes the transition in the areas of metastabilities ( * c      and 1 c     ) due to the generation of collective modes of defects (autosolitary waves of strain localization transforming into the blow- up dissipative structures), which provides finally singular damage kinetics on the set of spatial lengths , 1, 2,... H c L kL k K   . Different nature of these singularities, related to spatial (geometrical) basis for stress intensity factor and temporal one for damage localization, leads to the variety of transition scenario to failure in wide range of load intensity. Experimental study of dynamic crack instability in PMMA revealing the transition from steady to branching regimes of crack dynamics with qualitative changes of fracture surface morphology [19, 20], Fig.3; the “resonance” excitation of selected blow-up modes (equal size of mirror-like damage localization zones in different spall cross-sections of shocked PMMA rod [21,22]), the limit case of resonance damage localization under the “failure wave” initiation [23, 24], temporal and spatial scaling under dynamic fragmentation [25] supported the competitive role of two mentioned T